<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">o1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b1</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>;<br/><b>let </b><font color="Maroon">b2</font> be    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> holds <br/> ex <font color="Olive">a1'</font>, <font color="Olive">b1'</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">Fk</font> being   <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1'</font>,<font color="Olive">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">Fk</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Olive">Fk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> ) ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="funct_1.html#NM1">Function</a>] <b>means </b>(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> 1 ) ) );<br/>
<a NAME="E3:20"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">q</font> being   <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="funct_1.html#K1">.</a> 1 );<br/>
<a NAME="E2:20_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/> ex <font color="Olive">x</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:20_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E4:20_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E5:20_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p3">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">Gk</font> = <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:20_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:20_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gk</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">q1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span>, <font color="Maroon">q2</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E11:20_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120">FINSEQ_2:120</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1 as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> <b>by </b><i>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E13:20_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:20_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">p</font>
;<br/>

<b>thus </b><a NAME="E7:20_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E8:20_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:20_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E11:20_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">p</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<b>thus </b><a NAME="E12:20_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E13:20_1"/>
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1">POLYNOM2:sch 1</a>();<br/></i>
<a NAME="E7:20"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E8:20"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E10:20"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> holds <br/> ex <font color="Olive">a1'</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> ex <font color="Olive">Fk</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> st <br/>( <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> &amp; ( for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Olive">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Gi</font> being    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:20_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:20_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E5:20_2"/><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gi</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Gi</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:20_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:20_2"/><b>then </b><i><font color="Green">E76</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> &amp; ( for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font></span>)</span> holds <br/> ex <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_1.html#K1">.</a> 1 ) ) )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Fk</font> = <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">a1'</font>
;<br/>

<b>take </b>
<font color="Maroon">Fk</font>
;<br/>

<a NAME="E10:20_2"/><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<b>thus </b><a NAME="E11:20_2"/>
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 ;<br/>

<b>thus </b><a NAME="E12:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>by </b><i>, , , <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>

<b>thus </b> <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E14:20_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>assume </b><a NAME="E15:20_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E17:20_2"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E18:20_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Gi</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">a1''</font>
;<br/>

<a NAME="E19:20_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>hence </b><a NAME="E20:20_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1''</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>

<b>consider </b><font color="Maroon">p</font> being    <a href="finseq_2.html#M2">Element</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:20_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>and </b><br/><a NAME="E23:20_2"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1
 <b>by </b><i>, , </i>;<br/>
<a NAME="E24:20_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 
<b>by </b><i>, , , , <a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>thus </b><font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> = 
<font color="Maroon">Fk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
<b>by </b><i><a class="txt" href="polynom6.html#E3:20"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:20"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="polynom1.html#K20">divisors</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">FG</font> =  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font> as    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">bb</font> = <font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font> ;<br/>
<a NAME="E14:20"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E15:20"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E16:20"/><i><font color="Green">E133</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green">E134</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>
;<br/>
<a NAME="E18:20"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E19:20"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_4"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_4"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:20_4"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:20_4"/><i><font color="Green">E142</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E5:20_4"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E6:20_4"/><i><font color="Green">E144</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>
<a NAME="E7:20_4"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E9:20_4"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font>
 <b>and </b><br/><a NAME="E11:20_4"/><i><font color="Green">E150</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_4"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">a1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">a1''</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">a2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b2'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Gk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:20_4"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">Gk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E15:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a2'</font>,<font color="Maroon">b2'</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E16:20_4"/><i><font color="Green">E179</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Gk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E17:20_4"/>
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gk</font> holds <br/> ex <font color="Olive">a2''</font>, <font color="Olive">b2''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a2''</font>,<font color="Olive">b2''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Gk</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a2''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b2'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b2''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E35</font></i></a></i>;<br/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom6.html#E3:20"><i><font color="Green">E37</font></i></a>, , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E19:20_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D2">CARD_3:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:20"/><b>then </b><i><font color="Green">E180</font></i>: 
 <a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<a NAME="E21:20"/><b>then </b><i><font color="Green">E181</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T31">POLYNOM1:31</a></i>;<br/>
<a NAME="E22:20"/><b>then </b><i><font color="Green">E182</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FG</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bb</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:20"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FG</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bb</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">FG</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">p</font>,<span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K10">-'</a> <font color="Olive">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20_5"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">FG</font>
 <b>and </b><br/><a NAME="E2:20_5"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">bb</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:20_5"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">G</font>
 <b>and </b><br/><a NAME="E5:20_5"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:20_5"/><i><font color="Green">E188</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<a NAME="E8:20_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then consider </b><font color="Maroon">i'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j'</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:20_5"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E11:20_5"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">j'</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span>
 <b>and </b><br/><a NAME="E12:20_5"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5">Card</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p4">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j'</font>
 <b>and </b><br/><a NAME="E13:20_5"/><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T32">POLYNOM1:32</a></i>;<br/>
<a NAME="E14:20_5"/>
(  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">G</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="polynom1.html#K5">Card</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5">Card</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">i'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom3.html#T16">POLYNOM3:16</a></i>;<br/>
<a NAME="E15:20_5"/><b>then </b><i><font color="Green">E193</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j'</font> )
 
<b>by </b><i>, , <a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom6.html#E2:20"><i><font color="Green">E36</font></i></a>, , , , <a class="ref" href="polynom3.html#T22">POLYNOM3:22</a></i>;<br/>
<b>consider </b><font color="Maroon">a1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">b1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Gi</font> being    <a href="finseq_1.html#M2">FinSequence</a> of 2 <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <span class="p2">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E17:20_5"/><i><font color="Green">E194</font></i>: 
<font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E18:20_5"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b1</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E19:20_5"/><i><font color="Green">E196</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E20:20_5"/><i><font color="Green">E197</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gi</font> holds <br/> ex <font color="Olive">a1''</font>, <font color="Olive">b1''</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Olive">a1''</font>,<font color="Olive">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> &amp; <font color="Maroon">Gi</font> <a href="matrlin.html#K1">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E21:20_5"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Gi</font>
 
<b>by </b><i>, <a class="txt" href="polynom6.html#E1:20"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font>, <font color="Maroon">b1''</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E23:20_5"/><i><font color="Green">E199</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>and </b><br/><a NAME="E24:20_5"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">Gi</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</font></span>)</span></span><a href="polynom3.html#K2">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fa1</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font>, <font color="Maroon">Fk</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="polynom1.html#K14">Bags</a> <span class="p1">(<span class="default"><font color="Maroon">o1</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">o2</font></span>)</span><b> such that </b><br/><a NAME="E26:20_5"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E27:20_5"/><i><font color="Green">E202</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b1</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa1</font>
 <b>and </b><br/><a NAME="E28:20_5"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">Fk</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span>
 <b>and </b><br/><a NAME="E29:20_5"/><i><font color="Green">E203</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font> holds <br/> ex <font color="Olive">fa1'</font> being   <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">fa1'</font> &amp; <font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Olive">fa1'</font> )
 <b>by </b><i>, , , </i>;<br/>
<a NAME="E30:20_5"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">Fk</font>
 
<b>by </b><i>, , , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">fa1'</font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font><b> such that </b><br/><a NAME="E32:20_5"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K20">divisors</a> <font color="Maroon">b2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa1'</font>
 <b>and </b><br/><a NAME="E33:20_5"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">Fk</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fa1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">fa1'</font>
 <b>by </b><i/>;<br/>
<a NAME="E34:20_5"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">a1'</font> <a href="pboole.html#R6">=</a> <font color="Maroon">fa1</font>
 
<b>by </b><i>, , , , <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>
<a NAME="E35:20_5"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21">decomp</a> <font color="Maroon">b2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom6.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E36:20_5"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">a1''</font> <a href="pboole.html#R6">=</a> <font color="Maroon">fa1'</font>
 
<b>by </b><i>, , <a class="ref" href="polynom1.html#T74">POLYNOM1:74</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1a1</font> = <font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1'</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o1</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b2a1</font> = <font color="Maroon">b2</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1''</font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">o2</font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E39:20_5"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>then </b><i><font color="Green">E210</font></i>: <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">Fk</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="polynom6.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">a1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">a1''</font>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E41:20_5"/><i><font color="Green">E212</font></i>: 
<span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<font color="Maroon">b1'</font></span><a href="polynom3.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1'</font>,<span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1'</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , , , , <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<a NAME="E42:20_5"/><i><font color="Green">E213</font></i>: 
<span class="p1"><a href="polynom3.html#K2">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<font color="Maroon">b1''</font></span><a href="polynom3.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">a1''</font>,<span class="p2">(<span class="default"><font color="Maroon">b2</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">a1''</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>
<i><font color="Green">E214</font></i>: <font color="Maroon">bb</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">p</font> = 
<font color="Maroon">b1a1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2a1</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2a1</font>
<b>by </b><i>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>
<br/>.= 
<font color="Maroon">b1'</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b1''</font>
<b>by </b><i>, <a class="ref" href="group_7.html#T2">GROUP_7:2</a></i>
;<br/>
<b>thus </b><font color="Maroon">FG</font> <a href="matrlin.html#K1">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="txt" href="polynom6.html#E3:20"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">Gi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><font color="Maroon">bb</font> <a href="polynom1.html#K10">-'</a> <font color="Maroon">p</font></span>)</span></span><a href="finseq_1.html#K10">*&gt;</a></span>
<b>by </b><i>, , , , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E24:20"/>
 <a href="polynom1.html#K21">decomp</a> <span class="p1">(<span class="default"><font color="Maroon">b1</font> <a href="polynom6.html#K1" target="_self">+^</a> <font color="Maroon">b2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="polynom6.html#T1" target="_self">Th1</a>, <a class="ref" href="polynom1.html#D19">POLYNOM1:def 19</a></i>;<br/>


</div>
